Complete orbits and their partition

Goal and scope

The distinct orbits are {0,1,2} and {3}; same-orbit is an equivalence relation.

The same complete six-element/four-point action, using every g rather than bounded generator steps.

Givens and symbols

• calc

• G1 — G={e,r,r2,s,rs,r2s}, encoded respectively (0,0),(1,0),(2,0),(0,1),(1,1),(2,1), with (i,j)(k,l)=(i+(-1)^j*k mod3,j+l mod2). Points X={0,1,2,3}; (i,j) acts on x<3 by i+(-1)^j*x mod3 and fixes 3.

• G — six-element group carrier
• r,s — labels with r^3=e,s^2=e,sr=r^2s
• G\cdot x — all group images of a specified point

Rules and sources

• calc · prerequisite — Assume exact arithmetic modulo 3 and 2, extensional finite-set equality, and definitions of groups/actions; this operation is not pre-assumed to be a group.
• mko_definition · cited — Start with a fully qualified finite group action and enumerate G*x={g*x:g∈G} using every group element. The relation x~y exactly when some g sends x to y is an equivalence relation. Orbits are equal or disjoint and cover the carrier. One action image, finitely sampled generator steps and a stabilizer are not the whole orbit. · mko-finite-group-orbit

Main derivation

G=\{e,r,r^2,s,rs,r^2s\}

\begin{array}{c|cccccc}*&e&r&r^2&s&rs&r^2s\\\hline e&e&r&r^2&s&rs&r^2s\\r&r&r^2&e&rs&r^2s&s\\r^2&r^2&e&r&r^2s&s&rs\\s&s&r^2s&rs&e&r^2&r\\rs&rs&s&r^2s&r&e&r^2\\r^2s&r^2s&rs&s&r^2&r&e\\\end{array}

(i,j)(k,l)=(i+(-1)^jk,j+l)\quad(\bmod(3,2))

((i,j)(k,l))(m,n)=(i+(-1)^jk+(-1)^{j+l}m,j+l+n)=(i,j)((k,l)(m,n));\quad e=(0,0);\quad(i,j)^{-1}=(-(-1)^ji,j)

X=\{0,1,2,3\}

\begin{array}{c|cccc}\cdot&0&1&2&3\\\hline e&0&1&2&3\\r&1&2&0&3\\r^2&2&0&1&3\\s&0&2&1&3\\rs&1&0&2&3\\r^2s&2&1&0&3\\\end{array}

g=e,r,r^2,s,rs,r^2s

g\cdot0=0,1,2,0,1,2\Rightarrow G\cdot0=\{0,1,2\}

g\in G

g\cdot3=3\Rightarrow G\cdot3=\{3\}

x\sim y\iff\exists g:\ g\cdot x=y

e\cdot x=x;\quad g\cdot x=y\Rightarrow g^{-1}\cdot y=x;\quad h\cdot y=z\Rightarrow(hg)\cdot x=z

g\cdot1=1,2,0,2,0,1;\quad g\cdot2=2,0,1,1,2,0

G\cdot1=G\cdot2=\{0,1,2\}

O=\{0,1,2\},\quad P=\{3\}

O\cap P=\varnothing,\quad O\cup P=X,\quad X/G=\{O,P\}

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, S04, S05, S07 — This partitions points and is not a collection of stabilizers or group cosets.
• S07 · premise_audit · G1, S03, S04, S06 — Different base points may have equal orbit sets, which must not be counted as distinct orbits.
• S06 · premise_audit · G1, S02, S03 — The combined witness is hg because g acts first and then h.
• S05 · premise_audit · G1, S03 — The orbit is nonempty since e*3=3; a fixed point still has an orbit.
• S04 · premise_audit · G1, S03 — Taking only e or the single image r*0=1 is not the whole orbit.
• S03 · premise_audit · G1, S02 — Every group-point pair has one value; the formula covers all 144 triples, not merely permutation rows.
• S02 · premise_audit · G1, S01 — The sign depends only on j+l modulo 2 since (-1)^2=1; the full inverse list is e,r^2,r,s,rs,r^2s.
• S01 · premise_audit · G1 — The six labels bind the six pairs; r^2 is not an additional seventh element.

Error controls and negative controls

Same group table

Bind every product, inverse and coset to this complete Cayley table.

Complete enumeration

Every universal finite property needs the full carrier or a universal formula, not a proper sample.

• r*0=1 is the orbit. → reject — The orbit is the full image set {0,1,2}, not point 1.
• Point 3 has no orbit because it is fixed. → reject — Its orbit is exactly {3}.
• The orbits of 0 and 1 are disjoint. → reject — Both are {0,1,2} and are equal.

Result and limitations

The distinct orbits are {0,1,2} and {3}; same-orbit is an equivalence relation.

Verification and source binding

0 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: 15c74347343414557ec306e3bda093c9fb53df12da58f4a045f551f1a6c45b11
• original source SHA-256: 15c74347343414557ec306e3bda093c9fb53df12da58f4a045f551f1a6c45b11
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

Same-source full JSON (formulas, ASTs, controls and revision log)